In this paper, we generalize Colding and Minicozzi's work [5] on the stability of self-shrinkers in the hypersurface case to higher codimensional cases. The first and second variation formulae of the F -functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in [2] are unstable.

Regular shrinkers describe blow-up limits of a finite-time singularity of the motion by curvature of planar network of curves. This follows from Huisken's monotonicity formula. In this paper, we show that there is only one regular shrinker with 2 closed regions. This regular shrinker is the Cisgeminate eye. Moreover, we find some degenerate regular shrinkers with 2 closed regions.

In this paper, we generalize Colding and Minicozzi's work [5] on the stability of self-shrinkers in the hypersurface case to higher codimensional cases. The first and second variation formulae of the F -functional are derived and an equivalent condition to the stability in general codimension is found. Moreover, we show that the closed Lagrangian self-shrinkers given by Anciaux in [2] are unstable.

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